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We say that two sets are disjoint That is, T discrete is the collection of all subsets of X. Product Topology 6 6. If $\tau$ is the discrete topology on the real numbers, find the closure of $(a,b)$ Here is the solution from the back of my book: Since the discrete topology contains all subsets of $\Bbb{R}$, every subset of $\Bbb{R}$ is both open and closed. Quotient Topology â¦ Open sets Open sets are among the most important subsets of R. A collection of open sets is called a topology, and any property (such as â¦ Let Xbe any nonempty set. For example, the set of integers is discrete on the real line. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. Therefore, the closure of $(a,b)$ is â¦ TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by ââ©.â Aâ© B is the set of elements which belong to both sets A and B. $\endgroup$ â â¦ discrete:= P(X). Then consider it as a topological space R* with the usual topology. 52 3. In nitude of Prime Numbers 6 5. Another example of an infinite discrete set is the set . If anything is to be continuous, it's the real number line. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. De ne T indiscrete:= f;;Xg. Homeomorphisms 16 10. A set is discrete in a larger topological space if every point has a neighborhood such that . Universitext. I think not, but the proof escapes me. Typically, a discrete set is either finite or countably infinite. Subspace Topology 7 7. In: A First Course in Discrete Dynamical Systems. The real number field â, with its usual topology and the operation of addition, forms a second-countable connected locally compact group called the additive group of the reals. The question is: is there a function f from R to R* whose initial topology on R is discrete? Then T discrete is called the discrete topology on X. Perhaps the most important infinite discrete group is the additive group â¤ of the integers (the infinite cyclic group). Product, Box, and Uniform Topologies 18 11. $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? The real number line $\mathbf R$ is the archetype of a continuum. Consider the real numbers R first as just a set with no structure. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Example 3.5. A Theorem of Volterra Vito 15 9. 5.1. I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? 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